Math 74 Midterm 2: Solutions
November 12, 2008
1. Let X and Y be sets.
(a) Show that P (X Y ) = P (X ) P (Y ).
(b) If |X | = n, |Y | = m, and |X Y | = , calculate |P (X ) P (Y )|.
(c) With the same numbers as in (b), calculate |Pk (X Y )|, where
0 k |X Y
MATH 74 MIDTERM 2
All theorem references refer to the numbers on the note sheet. 1. A = cfw_1 and B = cfw_2 will work. [More generally, any two sets with the property that neither is a subset of the other will work] 2. Z = cfw_(1, 1), (2, 2), (3, 3) will
MATH 74 MIDTERM 2
1. Short reponse: 20 points For any set S , let P (S ) denote the set of all subsets of S . 1. [4 pts] Give an example of sets A and B such that P (A B ) = P (A) P (B ). You do not need to prove that your sets A and B have this property;
NOTES ON FUNCTIONS These notes will cover some terminology regarding functions not included in Solows book. You should read Appendix A.2 in the book before reading these notes. Denition 1. We say that two functions f and g are equal if they have the same
NOTES ON GROUP THEORY The goal of these notes is to introduce some of the basic concepts which come up in Math 113. As with the analysis/topology material, the emphasis will be placed on studying examples of algebra proofs and the specic techniques that t
NOTES ON LIMITS In these notes we give the denition of the limit of a function ; the material on sequences is in Appendix D.2 of the book. Our goal is to see how to use this denition to prove some of the types of statements which you will see in Math 104.
NOTES ON REAL NUMBERS In these notes we will construct the set of real numbers. Why would we want to do this you may ask? Well, mathematicians want mathematics to be based on a solid foundation, such as set theory. Adopting this point of view then require
NOTES ON RELATIONS These notes introduce the notion of a relation and, more importantly, the notion of an equivalence relation. Denition 1. A relation R between two sets A and B is a subset of the Cartesian product A B . We say that a A and b B are relate
NOTES ON SET THEORY The purpose of these notes is to cover some set theory terminology not included in Solows book. You should read Appendix A.1 in the book before reading these notes. The symbol := means that the thing on the left is being dened as the t
NOTES ON POINT-SET TOPOLOGY These notes introduce some basic concepts in the eld known as topology. Here we will only be concerned with point-set topology, essentially the topology of the set of real numbers. Topology is a huge area of mathematics, and yo
MATH 74 MIDTERM 2 NOTE SHEET
Theorem 1. A function f : X Y is invertible if and only if it is a bijection. Let Nk = cfw_x N : 1 x k . Theorem 2. If X is a nite set, then #(X ) = k if and only if there is a bijection f : Nk X . Theorem 3. If X and Y are ni
MATH 74 MIDTERM 1
Short response [20 points] 1. Answer true or false. [2 points each] (a) If is a binary operation on a set S , and is not commutative, then for any a and b in S , it must be that a b = b a. (b) The formula a b = a b + sin ab 2 , a, b Z,
d
MATH 74 MIDTERM 1
1(a). False. (No matter what is, and no matter what a S is, the equation a a = a a is always true, for example.) 1(b). True. For any integer k , the real number sin( k ) is an integer: it is either 0, 1, or 1 2 depending on whether k is
MATH 74 HOMEWORK 9 (DUE WEDNESDAY NOVEMBER 7)
Exercises 1,2. Let A = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1. Let X denote the set of subsets of A that have an odd number of elements. Let Y denote the set of subsets of A that have an even number of elements.
MATH 74 HOMEWORK 10 (DUE WEDNESDAY NOVEMBER 28)
1(a). We rst prove that if 5 | a, then 5 | a2 . Suppose 5 | a. This means that there is q Z with a = 5q . Squaring both sides we see a2 = 25q 2 = 5(q 2 ), so that 5 | a2 . We now prove that if 5 | a2 , then
MATH 74 HOMEWORK 10 (DUE WEDNESDAY NOVEMBER 28)
1(a). Problem 15.2 on page 198 of Eccles. 1(b). Give an example of an integer n with the property that the statement of 1(a), with n substituted for 5, becomes a false statement. Give an explicit example of
MATH 74 HOMEWORK 11
1. Since gcd(a1 , b1 ) = 1, a theorem from class gives us integers M and N with M a1 + N b1 = 1. Multiplying through by b2 we obtain M a1 b2 + N b1 b2 = b2 , and using the fact that a1 b2 = a2 b1 we get M a2 b1 + N b1 b2 = b2 . so that
MATH 74 HOMEWORK 11 (DUE FRIDAY DECEMBER 7)
Note the nonstandard due date. 1. Suppose that a1 , a2 , b1 , b2 are positive integers, that (1) that (2) and that (3) a1 b2 = a2 b1 . Prove that a1 = a2 and that b1 = b2 . 2(a). Compute, without proof, the set
MATH 74 FINAL EXAM THEOREM SHEET You may take for granted that + and are binary operations on Z satisfying the commutative, associative, and distributive laws. You thus do not need to pay attention to the placement of parentheses where these laws and the
FINAL EXAM COMMENTS AND SOLUTIONS
1. Comments 1. This was problem 5 on Homework 3. 2. This is a variation on problem 5 of Homework 3. (We have seen operations with this property before; see e.g. problem 2 on Homework 3.) 3. This is like problems 1-4 on Ho
MATH 74 FINAL EXAM [SOME TYPOS CORRECTED]
Short response. [6 pts each] 1. Give an example of an associative binary operation on R that is not commutative. 2. Give an example of a commutative binary operation # on R that is not associative. 3. Let S = cfw_
MATH 74 MIDTERM 1 SHEET
+ and are binary operations on R, and (A1) x + (y + z ) = (x + y ) + z for all x, y , and z in R (A2) x + y = y + x for all x and y in R (A3) 0 + x = x for all x in R (M1) (x y ) z = x (y z ) for all x, y , and z in R (M2) x y = y
Math 74 Final Exam: Solutions
December 16, 2008
1. (a) Let (X, d) be a metric space, and let (xn ) be a sequence in (X, d).
Dene what it means for (xn ) to be convergent.
(b) Use quantier negation to give a denition of (xn ) is not convergent.
(c) Pick yo
Math 74 Final Exam
December 16th, 2008
Name
SID
Question
1
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3
4
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Score
1
Possible
10
11
6
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61
1. (a) Let (X, d) be a metric space, and let (xn ) be a sequence in (X, d).
Dene what it means for (xn ) to be convergent.
(b) Use quantier neg
Math 74 Homework 4 Selected Solutions
September 22, 2008
1. Which of the following functions are injective? Which are surjective?
(You have to prove your answers.)
(a) f : Z Z, f (n) = n3 .
(b) g : R R, g (n) = n3 .
(c) h : Z N, h(n) = |n|. (Here N is the
05/09/2003 FRI 15:28 FAX 6434330 MOFFITT LIBRARY 001
J. Horowrl'z.
MATHEMATICS 74: FINAL EXAM
December 18, 2002
Follow all of the instructions carefully. Make sure to give reasons if they are asde for. Good luck!
Question I. (3 pts each)
Let A, B andCbe s
Section 6.6
Partial Orderings
Definition: Let R be a relation on A. Then R is a partial
order iff R is
reflexive
antisymmetric
and
transitive
(A, R) is called a partially ordered set or a poset.
_
Note: It is not required that two things be related und
Section 2: Reflexivity, Symmetry, and Transitivity
10.2.1
Definition: Let R be a binary relation on A. R is reflexive if for all x A, (x,x) R. (Equivalently, for all x e A, x R x.) R is symmetric if for all x,y A, (x,y) R implies (y,x) R. (Equivalently,
The Well-Ordering Principle
The well-ordering principle is a concept which is equivalent to mathematical induction. In your textbook, there is a proof for how the well-ordering principle implies
the validity of mathematical induction. However, because of
ax b mod m
Linear Congruences
Theorem 1. If (a, m) = 1, then the congruence ax b mod m phas exactly one solution
modulo m.
Constructive.
Solve the linear system
sa + tm = 1.
Then
sba + tbm = b.
So
sba b
(mod m)
gives the solution x = sb.
If u1 and u2 are